Alice and Brian are snails who live on a wall and can only travel
along the cracks. Alice wants to go to see Brian. How far is the
shortest route along the cracks? Is there more than one way to go?

Watch Your Feet

Age 7 to 11 Challenge Level:

Have you ever read a Winnie the Pooh poem called "Lines and
Squares" by A.A. Milne? It tells us how Christopher Robin plays a
game with himself as he walks along pavements, trying not to walk
on the "lines" (the edges of the paving stones).

Have you ever done this? I know I have! This investigation is
based on a similar idea - but here I want to walk on the
lines!

This is a picture of the path leading up to my front door from
the road:

I like to walk along the cracks of the paving stones, but not
the outside edge of the path itself because I may tread on the
grass.

The only way I can do it is by walking straight down the middle
of the path like this:

If the path were three paving stones wide instead of
just two it would look like this:

Remembering that I can only walk along the sides of the paving
stones and I mustn't tread on the outside edge, how many different
routes can you find for me to take? (By the way, you must not turn
back on yourself, and you must head towards the door or sides - so
you cannot walk towards the road on your journey.)

Do any of your routes have a repeating pattern?

Imagine now that the path is even wider, with
four paving stones:

What different routes can you find now?

Perhaps you could group them into those with a repeating pattern
and those without. Maybe you can find other ways to group the
routes.

If I could also take steps diagonally across a paving stone,
like this:

then there are even more possibilities. Try to find the new
routes yourself, starting with 3 paving stones width. Remember all
the same rules apply as before.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.